{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T15:53:26Z","timestamp":1786982006120,"version":"build-2736575974"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T00:00:00Z","timestamp":1757462400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T00:00:00Z","timestamp":1757462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-2052592"],"award-info":[{"award-number":["DMS-2052592"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Autonomous differential equation compartmental models hold broad utility in epidemiology and public health. However, these models typically cannot account explicitly for myriad factors that affect the trajectory of infectious diseases, with seasonal variations in host behavior and environmental conditions as noteworthy examples. Fortunately, using non-autonomous differential equation compartmental models can mitigate some of these deficiencies, as the inclusion of time-varying parameters can account for temporally varying factors. The inclusion of these temporally varying factors does come at a cost though, as many analysis techniques, such as the use of Poincar\u00e9 maps and Floquet theory, on non-autonomous differential equation compartmental models are typically only tractable numerically. Here, we illustrate a rare\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -strain generalized Susceptible-Infectious-Susceptible (SIS) compartmental model, with a general time-varying recovery rate, which features Floquet exponents that are algebraic expressions. We completely characterize the persistence and stability properties of our\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -strain generalized SIS model for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n\\ge 1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We also derive a closed-form solution in terms of elementary functions for the single-strain SIS model, which is capable of incorporating almost any infectious period distribution. Finally, to demonstrate the applicability of our work, we apply it to recent syphilis incidence data from the United States, utilizing Akaike Information Criteria and Forecast Skill Scores to inform on the model\u2019s goodness of fit relative to complexity and the model\u2019s capacity to predict future trends.\n                  <\/jats:p>","DOI":"10.1007\/s11538-025-01516-z","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T03:43:45Z","timestamp":1757475825000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Dynamics and Persistence of a Generalized Multi-strain SIS Model"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2484-4147","authenticated-orcid":false,"given":"Scott","family":"Greenhalgh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tabitha","family":"Henriquez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Frutschy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rebecah","family":"Leonard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,9,10]]},"reference":[{"key":"1516_CR1","doi-asserted-by":"publisher","first-page":"50","DOI":"10.5206\/mase\/9372","volume":"1","author":"M Al-arydah","year":"2020","unstructured":"Al-arydah M, Greenhalgh S, Munganga J, Smith R (2020) Applying the chemical-reaction definition of mass action to infectious disease modelling. Math Appl Sci Eng 1:50\u201364. https:\/\/doi.org\/10.5206\/mase\/9372","journal-title":"Math Appl Sci Eng"},{"key":"1516_CR2","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.1001\/jamadermatol.2022.3499","volume":"158","author":"EH Amerson","year":"2022","unstructured":"Amerson EH, Castillo Valladares HB, Leslie KS (2022) Resurgence of syphilis in the US\u2014USPSTF reaffirms screening guidelines. JAMA Dermatol 158:1241. https:\/\/doi.org\/10.1001\/jamadermatol.2022.3499","journal-title":"JAMA Dermatol"},{"key":"1516_CR3","doi-asserted-by":"publisher","first-page":"1549","DOI":"10.1038\/s41564-021-01000-z","volume":"6","author":"MA Beale","year":"2021","unstructured":"Beale MA, Marks M, Cole MJ et al (2021) Global phylogeny of Treponema pallidum lineages reveals recent expansion and spread of contemporary syphilis. Nat Microbiol 6:1549\u20131560. https:\/\/doi.org\/10.1038\/s41564-021-01000-z","journal-title":"Nat Microbiol"},{"key":"1516_CR4","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1080\/23737867.2018.1509026","volume":"5","author":"JC Blackwood","year":"2018","unstructured":"Blackwood JC, Childs LM (2018) An introduction to compartmental modeling for the budding infectious disease modeler. Lett Biomath 5:195\u2013221. https:\/\/doi.org\/10.1080\/23737867.2018.1509026","journal-title":"Lett Biomath"},{"key":"1516_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mbs.2018.12.015","volume":"309","author":"E Bonyah","year":"2019","unstructured":"Bonyah E, Khan MA, Okosun KO, G\u00f3mez-Aguilar JF (2019) Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control. Math Biosci 309:1\u201311. https:\/\/doi.org\/10.1016\/j.mbs.2018.12.015","journal-title":"Math Biosci"},{"key":"1516_CR6","doi-asserted-by":"crossref","unstructured":"Brauer F (2008) Compartmental Models in Epidemiology, pp 19\u201379","DOI":"10.1007\/978-3-540-78911-6_2"},{"key":"1516_CR7","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/j.idm.2017.04.001","volume":"2","author":"ARAR Brock","year":"2017","unstructured":"Brock ARAR, Ross JVJV, Greenhalgh S et al (2017) Modelling the impact of antimalarial quality on the transmission of sulfadoxine-pyrimethamine resistance in Plasmodium falciparum. Infect Dis Model 2:161\u2013187. https:\/\/doi.org\/10.1016\/j.idm.2017.04.001","journal-title":"Infect Dis Model"},{"key":"1516_CR8","doi-asserted-by":"publisher","first-page":"325","DOI":"10.2307\/2307403","volume":"60","author":"A Buckley","year":"1953","unstructured":"Buckley A (1953) The bernoulli differential equation. Am Math Mon 60:325","journal-title":"Am Math Mon"},{"key":"1516_CR9","unstructured":"Center for Disease Control and Prevention (2024) Sexually Transmitted Infections Surveillance, 2022"},{"key":"1516_CR10","doi-asserted-by":"publisher","first-page":"1272","DOI":"10.1007\/s11538-008-9299-0","volume":"70","author":"N Chitnis","year":"2008","unstructured":"Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol 70:1272\u20131296. https:\/\/doi.org\/10.1007\/s11538-008-9299-0","journal-title":"Bull Math Biol"},{"key":"1516_CR11","doi-asserted-by":"publisher","first-page":"873","DOI":"10.1098\/rsif.2009.0386","volume":"7","author":"O Diekmann","year":"2010","unstructured":"Diekmann O, Heesterbeek JAP, Roberts MG (2010) The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface 7:873\u2013885","journal-title":"J R Soc Interface"},{"key":"1516_CR12","first-page":"128","volume":"4","author":"A Fares","year":"2013","unstructured":"Fares A (2013) Factors influencing the seasonal patterns of infectious diseases. Int J Prev Med 4:128\u2013132","journal-title":"Int J Prev Med"},{"key":"1516_CR13","doi-asserted-by":"publisher","first-page":"16131","DOI":"10.3934\/mbe.2023720","volume":"20","author":"J Farrell","year":"2023","unstructured":"Farrell J, Spolyar O, Greenhalgh S (2023) The effect of screening on the health burden of chlamydia: an evaluation of compartmental models based on person-days of infection. Math Biosci Eng 20:16131\u201316147. https:\/\/doi.org\/10.3934\/mbe.2023720","journal-title":"Math Biosci Eng"},{"key":"1516_CR14","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/0025-5564(94)00069-C","volume":"128","author":"Z Feng","year":"1995","unstructured":"Feng Z, Thieme HR (1995) Recurrent outbreaks of childhood diseases revisited: the impact of isolation. Math Biosci 128:93\u2013130. https:\/\/doi.org\/10.1016\/0025-5564(94)00069-C","journal-title":"Math Biosci"},{"key":"1516_CR15","doi-asserted-by":"publisher","first-page":"e4383","DOI":"10.1371\/journal.pone.0004383","volume":"4","author":"AC Ghani","year":"2009","unstructured":"Ghani AC, Sutherland CJ, Riley EM et al (2009) Loss of population levels of immunity to malaria as a result of exposure-reducing interventions: consequences for interpretation of disease trends. PLoS ONE 4:e4383. https:\/\/doi.org\/10.1371\/journal.pone.0004383","journal-title":"PLoS ONE"},{"key":"1516_CR16","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/S0895-7177(97)00009-5","volume":"25","author":"D Greenhalgh","year":"1997","unstructured":"Greenhalgh D (1997) Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity. Math Comput Model 25:85\u2013107. https:\/\/doi.org\/10.1016\/S0895-7177(97)00009-5","journal-title":"Math Comput Model"},{"key":"1516_CR17","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1186\/s12889-019-8141-y","volume":"20","author":"S Greenhalgh","year":"2020","unstructured":"Greenhalgh S, Chandwani V (2020) Advocating an attack against severe malaria: a cost-effectiveness analysis. BMC Public Health 20:17. https:\/\/doi.org\/10.1186\/s12889-019-8141-y","journal-title":"BMC Public Health"},{"key":"1516_CR18","doi-asserted-by":"publisher","first-page":"1190","DOI":"10.1016\/j.idm.2023.11.007","volume":"8","author":"S Greenhalgh","year":"2023","unstructured":"Greenhalgh S, Dumas A (2023) A generalized ODE susceptible-infectious-susceptible compartmental model with potentially periodic behavior. Infect Dis Model 8:1190\u20131202. https:\/\/doi.org\/10.1016\/j.idm.2023.11.007","journal-title":"Infect Dis Model"},{"key":"1516_CR19","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1080\/023929021000008813","volume":"43","author":"D Greenhalgh","year":"2003","unstructured":"Greenhalgh D, Moneim IA (2003) Sirs epidemic model and simulations using different types of seasonal contact rate. Syst Anal Model Simul 43:573\u2013600. https:\/\/doi.org\/10.1080\/023929021000008813","journal-title":"Syst Anal Model Simul"},{"key":"1516_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.idm.2021.08.007","author":"S Greenhalgh","year":"2021","unstructured":"Greenhalgh S, Rozins C (2021) A generalized differential equation compartmental model of infectious disease transmission. Infect Dis Model. https:\/\/doi.org\/10.1016\/j.idm.2021.08.007","journal-title":"Infect Dis Model"},{"key":"1516_CR21","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.jtbi.2015.09.037","volume":"387","author":"S Greenhalgh","year":"2015","unstructured":"Greenhalgh S, Galvani APAP, Medlock J (2015a) Disease elimination and re-emergence in differential-equation models. J Theor Biol 387:174\u2013180. https:\/\/doi.org\/10.1016\/j.jtbi.2015.09.037","journal-title":"J Theor Biol"},{"key":"1516_CR22","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1097\/QAD.0000000000000550","volume":"29","author":"S Greenhalgh","year":"2015","unstructured":"Greenhalgh S, Ndeffo M, Galvani APAPAP, Parikh S (2015b) The epidemiological impact of HIV antiretroviral therapy on malaria in children. AIDS (London, England) 29:473\u2013482. https:\/\/doi.org\/10.1097\/QAD.0000000000000550","journal-title":"AIDS (London, England)"},{"key":"1516_CR23","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1097\/QAI.0000000000001783","volume":"79","author":"S Greenhalgh","year":"2018","unstructured":"Greenhalgh S, Hobbs CVCV, Parikh S (2018) Brief report: antimalarial benefit of HIV antiretroviral therapy in areas of low to moderate malaria transmission intensity. JAIDS J Acquir Immune Defic Syndr 79:249\u2013254. https:\/\/doi.org\/10.1097\/QAI.0000000000001783","journal-title":"JAIDS J Acquir Immune Defic Syndr"},{"key":"1516_CR24","unstructured":"Greenhalgh S (2022) mathguypi314\/GSISmodel: gSIS model"},{"key":"1516_CR25","doi-asserted-by":"crossref","unstructured":"Joshi HR, Lenhart S, Li MY, Wang L (2006) Optimal control methods applied to disease models. pp 187\u2013207","DOI":"10.1090\/conm\/410\/07728"},{"key":"1516_CR26","doi-asserted-by":"publisher","first-page":"473","DOI":"10.3934\/dcdsb.2002.2.473","volume":"2","author":"E Jung","year":"2002","unstructured":"Jung E, Lenhart S, Feng Z (2002) Optimal control of treatments in a two-strain tuberculosis model. Discrete Contin Dyn Syst B 2:473\u2013482. https:\/\/doi.org\/10.3934\/dcdsb.2002.2.473","journal-title":"Discrete Contin Dyn Syst B"},{"key":"1516_CR27","first-page":"57","volume":"53","author":"WO Kermack","year":"1991","unstructured":"Kermack WO, McKendrick AG (1991a) Contributions to the mathematical theory of epidemics\u2013II. The problem of endemicity. Bull Math Biol 53:57\u201387","journal-title":"Bull Math Biol"},{"key":"1516_CR28","first-page":"33","volume":"53","author":"WO Kermack","year":"1991","unstructured":"Kermack WO, McKendrick AG (1991b) Contributions to the mathematical theory of epidemics\u2013I. 1927. Bull Math Biol 53:33\u201355","journal-title":"Bull Math Biol"},{"key":"1516_CR29","first-page":"89","volume":"53","author":"WO Kermack","year":"1991","unstructured":"Kermack WO, McKendrick AG (1991c) Contributions to the mathematical theory of epidemics\u2013III. Further studies of the problem of endemicity. Bull Math Biol 53:89\u2013118","journal-title":"Bull Math Biol"},{"key":"1516_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2020\/1943410","volume":"2020","author":"A Kouidere","year":"2020","unstructured":"Kouidere A, Labzai A, Ferjouchia H et al (2020) A new mathematical modeling with optimal control strategy for the dynamics of population of diabetics and its complications with effect of behavioral factors. J Appl Math 2020:1\u201312. https:\/\/doi.org\/10.1155\/2020\/1943410","journal-title":"J Appl Math"},{"key":"1516_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3978-7","volume-title":"Elements of applied bifurcation theory","author":"YA Kuznetsov","year":"2004","unstructured":"Kuznetsov YA (2004) Elements of applied bifurcation theory. Springer, New York"},{"key":"1516_CR32","unstructured":"Levine R (2024) Syphilis Is a Public Health Priority"},{"key":"1516_CR33","unstructured":"Lloyd A (2017) Introduction to Epidemiological Modeling: Basic Models and Their Properties"},{"key":"1516_CR34","doi-asserted-by":"publisher","first-page":"2389","DOI":"10.1142\/S0218127401003486","volume":"11","author":"K Lust","year":"2001","unstructured":"Lust K (2001) Improved numerical floquet multipliers. Int J Bifurcat Chaos 11:2389\u20132410. https:\/\/doi.org\/10.1142\/S0218127401003486","journal-title":"Int J Bifurcat Chaos"},{"key":"1516_CR35","doi-asserted-by":"publisher","first-page":"161","DOI":"10.3934\/mbe.2006.3.161","volume":"3","author":"J Ma","year":"2006","unstructured":"Ma J, Ma Z (2006) Epidemic threshold conditions for seasonally forced SEIR models. Math Biosci Eng 3:161\u2013172. https:\/\/doi.org\/10.3934\/mbe.2006.3.161","journal-title":"Math Biosci Eng"},{"key":"1516_CR36","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.cam.2016.02.051","volume":"307","author":"A Mallela","year":"2016","unstructured":"Mallela A, Lenhart S, Vaidya NK (2016) HIV\u2013TB co-infection treatment: modeling and optimal control theory perspectives. J Comput Appl Math 307:143\u2013161. https:\/\/doi.org\/10.1016\/j.cam.2016.02.051","journal-title":"J Comput Appl Math"},{"key":"1516_CR37","doi-asserted-by":"publisher","first-page":"7499","DOI":"10.3390\/jcm11247499","volume":"11","author":"V Marchese","year":"2022","unstructured":"Marchese V, Tiecco G, Storti S et al (2022) Syphilis infections, reinfections and serological response in a large Italian sexually transmitted disease centre: a monocentric retrospective study. JCM 11:7499. https:\/\/doi.org\/10.3390\/jcm11247499","journal-title":"JCM"},{"key":"1516_CR38","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1080\/17513750802638712","volume":"3","author":"M Martcheva","year":"2009","unstructured":"Martcheva M (2009) A non-autonomous multi-strain SIS epidemic model. J Biol Dyn 3:235\u2013251. https:\/\/doi.org\/10.1080\/17513750802638712","journal-title":"J Biol Dyn"},{"key":"1516_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-7612-3","volume-title":"An Introduction to Mathematical Epidemiology","author":"M Martcheva","year":"2015","unstructured":"Martcheva M (2015) An Introduction to Mathematical Epidemiology. Springer, US"},{"key":"1516_CR40","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pntd.0003234","author":"MLN Mbah","year":"2014","unstructured":"Mbah MLN, Skrip L, Greenhalgh S et al (2014) Impact of Schistosoma mansoni on malaria transmission in sub-Saharan Africa. PLoS Negl Trop Dis. https:\/\/doi.org\/10.1371\/journal.pntd.0003234","journal-title":"PLoS Negl Trop Dis"},{"key":"1516_CR41","doi-asserted-by":"publisher","first-page":"591","DOI":"10.3934\/mbe.2005.2.591","volume":"2","author":"IA Moneim","year":"2005","unstructured":"Moneim IA, Greenhalgh D (2005) Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate. Math Biosci Eng 2:591\u2013611. https:\/\/doi.org\/10.3934\/mbe.2005.2.591","journal-title":"Math Biosci Eng"},{"key":"1516_CR42","doi-asserted-by":"publisher","first-page":"2417","DOI":"10.1175\/1520-0493(1988)116<2417:SSBOTM>2.0.CO;2","volume":"116","author":"AH Murphy","year":"1988","unstructured":"Murphy AH (1988) Skill scores based on the mean square error and their relationships to the correlation coefficient. Mon Wea Rev 116:2417\u20132424. https:\/\/doi.org\/10.1175\/1520-0493(1988)116%3c2417:SSBOTM%3e2.0.CO;2","journal-title":"Mon Wea Rev"},{"key":"1516_CR43","doi-asserted-by":"publisher","first-page":"113678","DOI":"10.1016\/j.chaos.2023.113678","volume":"173","author":"F Nill","year":"2023","unstructured":"Nill F (2023) Endemic oscillations for SARS-CoV-2 Omicron\u2014a SIRS model analysis. Chaos Solitons Fractals 173:113678. https:\/\/doi.org\/10.1016\/j.chaos.2023.113678","journal-title":"Chaos Solitons Fractals"},{"key":"1516_CR44","doi-asserted-by":"publisher","DOI":"10.1515\/9781400840915","volume-title":"A biologist\u2019s guide to mathematical modeling in ecology and evolution","author":"SP Otto","year":"2007","unstructured":"Otto SP, Day T (2007) A biologist\u2019s guide to mathematical modeling in ecology and evolution. Princeton University Press"},{"key":"1516_CR45","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/0022-247X(75)90156-0","volume":"51","author":"KJ Palmer","year":"1975","unstructured":"Palmer KJ (1975) Linearization near an integral manifold. J Math Anal Appl 51:243\u2013255. https:\/\/doi.org\/10.1016\/0022-247X(75)90156-0","journal-title":"J Math Anal Appl"},{"key":"1516_CR46","doi-asserted-by":"publisher","DOI":"10.18637\/jss.v102.i05","author":"A Puy","year":"2022","unstructured":"Puy A, Piano SL, Saltelli A, Levin SA (2022) sensobol: an R package to compute variance-based sensitivity indices. J Stat Soft. https:\/\/doi.org\/10.18637\/jss.v102.i05","journal-title":"J Stat Soft"},{"key":"1516_CR47","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1177\/0956462419886772","volume":"31","author":"X Tao","year":"2020","unstructured":"Tao X, Ghanem KG, Page KR et al (2020) Risk factors predictive of sexually transmitted infection diagnosis in young compared to older patients attending sexually transmitted diseases clinics. Int J STD AIDS 31:142\u2013149. https:\/\/doi.org\/10.1177\/0956462419886772","journal-title":"Int J STD AIDS"},{"key":"1516_CR48","doi-asserted-by":"publisher","first-page":"1128","DOI":"10.1080\/00036811.2014.918606","volume":"94","author":"JP Tian","year":"2015","unstructured":"Tian JP, Wang J (2015) Some results in Floquet theory, with application to periodic epidemic models. Appl Anal 94:1128\u20131152. https:\/\/doi.org\/10.1080\/00036811.2014.918606","journal-title":"Appl Anal"},{"key":"1516_CR49","first-page":"93","volume-title":"Da Silva FAB, Carels N, Trindade Dos Santos M","author":"M Tosin","year":"2020","unstructured":"Tosin M, C\u00f4rtes AMA, Cunha A (2020) A Tutorial on Sobol\u2019 Global Sensitivity Analysis Applied to Biological Models. In: Lopes FJP (ed) Da Silva FAB, Carels N, Trindade Dos Santos M. Networks in Systems Biology. Springer International Publishing, Cham, pp 93\u2013118"},{"key":"1516_CR50","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.ijid.2018.04.4180","volume":"73","author":"S Tsuzuki","year":"2018","unstructured":"Tsuzuki S, Yamaguchi T, Nishiura H (2018) Time series transition of basic reproduction number of Syphilis in Japan. Int J Infect Dis 73:337. https:\/\/doi.org\/10.1016\/j.ijid.2018.04.4180","journal-title":"Int J Infect Dis"},{"key":"1516_CR51","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s10884-008-9111-8","volume":"20","author":"W Wang","year":"2008","unstructured":"Wang W, Zhao X-Q (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20:699\u2013717. https:\/\/doi.org\/10.1007\/s10884-008-9111-8","journal-title":"J Dyn Differ Equ"}],"container-title":["Bulletin of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-025-01516-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11538-025-01516-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-025-01516-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T08:17:48Z","timestamp":1758788268000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11538-025-01516-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,10]]},"references-count":51,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1516"],"URL":"https:\/\/doi.org\/10.1007\/s11538-025-01516-z","relation":{},"ISSN":["0092-8240","1522-9602"],"issn-type":[{"value":"0092-8240","type":"print"},{"value":"1522-9602","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,10]]},"assertion":[{"value":"8 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"147"}}